10, 1000, 10239, what do they all have common? They all have the number zero. But what is the purpose and origin of this number zero? What are implications of the number zero? An infinity means something without limit. But what is it actually? Are all Infinities the same?
Zero (0)
Zero has not always been around, with its first know English use in 1598. The word zero came from the French word zéro, which in turn came from Venetian zero which came via Italian word Zefiro which came from the Arabic word Safira, meaning it was empty. But why have zero?
In ancient times, humans used different ways to keep track of items, eventually developing numbers. This was kind of obvious to people at the time, to have a system to tell others how many sheep they had. But they had no use for the number zero. If someone asked you how many sheep you had, and you had none, you could just tell them you had no sheep. Why say zero sheep?
But we use Base 10 in counting and this means without zero, we had no way to mark out the places of tens. When maths had developed further from simply counting, we needed a way to clearly write out the places of tens. We couldn't just write 1 for 10 as it also means 1. In 976 AD the Persian encyclopedist Muhammad ibn Ahmad al-Khwarizmi, wrote in his "Keys of the Sciences" that a circle would be used in place for the places of tens and it was called Safira, meaning it was empty.
Now there was a clear way to write out the places of tens. Instead of writing to your military adviser that you had 2 9 men, you could write 2009, with the 0 representing tens and hundreds position.
Of course with the creation of zero as a number means that there had to be certain rules designed for it. The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe), written in 628 AD. They were:
The sum of zero and a negative number is negative.
The sum of zero and a positive number is positive.
The sum of zero and zero is zero.
The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
A positive or negative number when divided by zero is a fraction with the zero as denominator.
Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
Zero divided by zero is zero.
The last rule is not applicable now as mathematicians have determined that anything divided by zero is undetermined but zero divided by anything non-zero is equal to zero. It would be interesting to note that the product of zero numbers is 1.
Infinity
Infinity is a concept which usually means without a limit and the word infinity is derived from the Latin word infinitas which means "the state of being without finish" which can be translated to mean boundless or endless. Although this meanings for infinity suggest infinity as a state rather than a number, in mathematics it is usually treated as a number. However they are not treated as the same sort of number as real numbers.
Problems with Zero
The main problem with zero is division by zero. As stated above, anything divided by zero is undetermined. Now, we will attempt to place this fact.
Now some people might say that something divided by 0 is infinity. Or x/0=infinity. Now what exactly is division? First what is multiplication? We have used this two terms for so long yet we have not actually thought of what they actually do. Now multiplication is just glorified addition. What that means is if we consider 2+2+2, it is equivalent to 2x3. So to work out AxB, its basically adding A to itself B number of times. What about division? Division is just glorified subtraction. What that means is to work out 6 divided by 2, we do a series of subtractions. Or:
6-2=4
4-2=2
2-2=0 This took 3 times so 6/2 is 3
In other words, division of A/B=C is A-B C amounts of time until A is reduced to 0.
Keeping this in mind, we can work out something like 2/0. We would have a series of subtractions as such:
2-0=2
2-0=2
2-0=2 and this goes on forever.
So some people might argue this means that dividing by zero gives infinity. But this isn't true.
Take a graph and draw the graph y=1/x as x reaches 0. Some people say that this is equal to infinity. But if you come into y=1/x as x reaches 0 through the positive side, this is true but if you come in from the negative side, it is negative infinity. If you came in from a complex number, than it approaches 0 in different ways. Thus anything divided by zero is undefined.
Problems with Infinities
I'm just gonna highlight one paradox involving zero. Imagine you are two meters away from your best friend and you want to go over and hi-five his hand. Initially you are 2m away from each other. If he doesn't move and you move, to reach him you need to travel half the distance of the 2m, which is 1m. After travelling that 1m, you have to travel another half of 1m to reach him. You keep going and you will realise that even though the distance between you two is getting smaller and smaller, you will never actually reach him as you can keep adding halves of halves of halves forever and never reach the distance of 2. In other words, 1+1/2+1/4+1/8+1/16..... will never reach 2.
This is known as Zeno's Paradox and it cant be true. I mean, common sense dictates that you will eventually reach your friend. So how do you solve this?
Using a little bit of maths, you can make the sum converge. Supposing 1+1/2+1/4....=S,
2S= 2+1+1/2+1/4.... which you will notice that the original sum is reproduced behind the two so
2S=2+S Moving over the s you get
S=2 which produces the 2 you intend to get.
But what does this prove to show? Well first thing you can see is that infinity are not all the same size. If you consider 1+1/2+1/3... instead, it wouldn't produce a finite limit, giving a 'bigger' infinity.
One thing to note after all this, in the periodic table the last row is listed as 0 and Mr Din says that all the rows are listed using Roman Numerals. 0 isn't a Roman Numeral and there is no Roman Numeral for 0.
Zero (0)
Zero has not always been around, with its first know English use in 1598. The word zero came from the French word zéro, which in turn came from Venetian zero which came via Italian word Zefiro which came from the Arabic word Safira, meaning it was empty. But why have zero?
In ancient times, humans used different ways to keep track of items, eventually developing numbers. This was kind of obvious to people at the time, to have a system to tell others how many sheep they had. But they had no use for the number zero. If someone asked you how many sheep you had, and you had none, you could just tell them you had no sheep. Why say zero sheep?
But we use Base 10 in counting and this means without zero, we had no way to mark out the places of tens. When maths had developed further from simply counting, we needed a way to clearly write out the places of tens. We couldn't just write 1 for 10 as it also means 1. In 976 AD the Persian encyclopedist Muhammad ibn Ahmad al-Khwarizmi, wrote in his "Keys of the Sciences" that a circle would be used in place for the places of tens and it was called Safira, meaning it was empty.
Now there was a clear way to write out the places of tens. Instead of writing to your military adviser that you had 2 9 men, you could write 2009, with the 0 representing tens and hundreds position.
Of course with the creation of zero as a number means that there had to be certain rules designed for it. The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe), written in 628 AD. They were:
- The sum of zero and a negative number is negative.
- The sum of zero and a positive number is positive.
- The sum of zero and zero is zero.
- The sum of a positive and a negative is their difference; or, if their absolute values are equal, zero.
- A positive or negative number when divided by zero is a fraction with the zero as denominator.
- Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
- Zero divided by zero is zero.
The last rule is not applicable now as mathematicians have determined that anything divided by zero is undetermined but zero divided by anything non-zero is equal to zero. It would be interesting to note that the product of zero numbers is 1.Infinity
Infinity is a concept which usually means without a limit and the word infinity is derived from the Latin word infinitas which means "the state of being without finish" which can be translated to mean boundless or endless. Although this meanings for infinity suggest infinity as a state rather than a number, in mathematics it is usually treated as a number. However they are not treated as the same sort of number as real numbers.
Problems with Zero
The main problem with zero is division by zero. As stated above, anything divided by zero is undetermined. Now, we will attempt to place this fact.
Now some people might say that something divided by 0 is infinity. Or x/0=infinity. Now what exactly is division? First what is multiplication? We have used this two terms for so long yet we have not actually thought of what they actually do. Now multiplication is just glorified addition. What that means is if we consider 2+2+2, it is equivalent to 2x3. So to work out AxB, its basically adding A to itself B number of times. What about division? Division is just glorified subtraction. What that means is to work out 6 divided by 2, we do a series of subtractions. Or:
6-2=4
4-2=2
2-2=0 This took 3 times so 6/2 is 3
In other words, division of A/B=C is A-B C amounts of time until A is reduced to 0.
Keeping this in mind, we can work out something like 2/0. We would have a series of subtractions as such:
2-0=2
2-0=2
2-0=2 and this goes on forever.
So some people might argue this means that dividing by zero gives infinity. But this isn't true.
Take a graph and draw the graph y=1/x as x reaches 0. Some people say that this is equal to infinity. But if you come into y=1/x as x reaches 0 through the positive side, this is true but if you come in from the negative side, it is negative infinity. If you came in from a complex number, than it approaches 0 in different ways. Thus anything divided by zero is undefined.
Problems with Infinities
I'm just gonna highlight one paradox involving zero. Imagine you are two meters away from your best friend and you want to go over and hi-five his hand. Initially you are 2m away from each other. If he doesn't move and you move, to reach him you need to travel half the distance of the 2m, which is 1m. After travelling that 1m, you have to travel another half of 1m to reach him. You keep going and you will realise that even though the distance between you two is getting smaller and smaller, you will never actually reach him as you can keep adding halves of halves of halves forever and never reach the distance of 2. In other words, 1+1/2+1/4+1/8+1/16..... will never reach 2.
This is known as Zeno's Paradox and it cant be true. I mean, common sense dictates that you will eventually reach your friend. So how do you solve this?
Using a little bit of maths, you can make the sum converge. Supposing 1+1/2+1/4....=S,
2S= 2+1+1/2+1/4.... which you will notice that the original sum is reproduced behind the two so
2S=2+S Moving over the s you get
S=2 which produces the 2 you intend to get.
But what does this prove to show? Well first thing you can see is that infinity are not all the same size. If you consider 1+1/2+1/3... instead, it wouldn't produce a finite limit, giving a 'bigger' infinity.
One thing to note after all this, in the periodic table the last row is listed as 0 and Mr Din says that all the rows are listed using Roman Numerals. 0 isn't a Roman Numeral and there is no Roman Numeral for 0.